
where the scattering rate \(W^{\chi \chi ^{\prime }}_{\mathbf {k},\mathbf {k}^{\prime }}\)  is given by [1]}
\(W^{\chi \chi ^{\prime }}_{\mathbf {k},\mathbf {k}^{\prime }} = \frac{2\pi }{\hbar } \frac{n}{\mathcal {V}} |\langle \psi ^{\chi ^{\prime }}_{\mathbf {k}^{\prime }}|U^{\chi \chi ^{\prime }}_{\mathbf {k}\mathbf {k}^{\prime }}|\psi ^\chi _\mathbf {k}\rangle |^2 \delta (\epsilon ^{\chi ^{\prime }}_{\mathbf {k}^{\prime }}-\epsilon _F)\) 
